Integrand size = 15, antiderivative size = 125 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \]
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Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {a d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {a \cosh (c+d x)}{b^2 (a+b x)} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a \cosh (c+d x)}{b (a+b x)^2}+\frac {\cosh (c+d x)}{b (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{a+b x} \, dx}{b}-\frac {a \int \frac {\cosh (c+d x)}{(a+b x)^2} \, dx}{b} \\ & = \frac {a \cosh (c+d x)}{b^2 (a+b x)}-\frac {(a d) \int \frac {\sinh (c+d x)}{a+b x} \, dx}{b^2}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b} \\ & = \frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {\left (a d \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2}-\frac {\left (a d \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^2} \\ & = \frac {a \cosh (c+d x)}{b^2 (a+b x)}+\frac {\cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {a d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^2} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {\frac {a b \cosh (c+d x)}{a+b x}+\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \cosh \left (c-\frac {a d}{b}\right )-a d \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-a d \cosh \left (c-\frac {a d}{b}\right )+b \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(129)=258\).
Time = 0.20 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.35
method | result | size |
risch | \(\frac {d \,{\mathrm e}^{-d x -c} a}{2 b^{2} \left (d x b +d a \right )}-\frac {d \,{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a}{2 b^{3}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 b^{2}}+\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a b d x +{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{2} d -{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) b^{2} x -{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a b +{\mathrm e}^{d x +c} a b}{2 b^{3} \left (b x +a \right )}\) | \(294\) |
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Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.60 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\frac {2 \, a b \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (a^{2} d - a b + {\left (a b d - b^{2}\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{2} d + a b + {\left (a b d + b^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{4} x + a b^{3}\right )}} \]
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\[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {1}{2} \, {\left (a {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}} - \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b^{3}}\right )} + \frac {\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}}{b d} + \frac {2 \, \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{2} d}\right )} d + {\left (\frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}}\right )} \cosh \left (d x + c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (129) = 258\).
Time = 0.31 (sec) , antiderivative size = 994, normalized size of antiderivative = 7.95 \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=-\frac {{\left ({\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b c d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + a^{2} d^{3} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + a b c d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a^{2} d^{3} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (\frac {b c - a d}{b}\right )} - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} + b^{2} c d {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} {\rm Ei}\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) e^{\left (-\frac {b c - a d}{b}\right )} - a b d^{2} e^{\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )} - a b d^{2} e^{\left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )}\right )} b}{2 \, {\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]
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Timed out. \[ \int \frac {x \cosh (c+d x)}{(a+b x)^2} \, dx=\int \frac {x\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
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